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The problem is asking to find the equation of the tangent line to the graph of the function:

\[
f(x) = 3xe^x
\]

at the point \((a, f(a))\) where \(a = 1\).

### Steps to find the tangent line equation:

1. **Function and Derivative:**
   The equation of a tangent line is given by:
   \[
   y - f(a) = f'(a)(x - a)
   \]
   So, we need to find:
   - \(f(1)\) (the value of the function at \(x = 1\)).
   - \(f'(x)\) (the derivative of the function to get the slope).
   - \(f'(1)\) (the value of the derivative at \(x = 1\)).

2. **Find \(f(x)\):**
   \[
   f(x) = 3xe^x
   \]
   We substitute \(x = 1\) into the function:
   \[
   f(1) = 3 \cdot 1 \cdot e^1 = 3e
   \]

3. **Find \(f'(x)\) (the derivative):**
   To differentiate \(f(x) = 3xe^x\), we apply the product rule:
   \[
   f'(x) = 3 \cdot \frac{d}{dx}(xe^x) = 3 \left( e^x + x e^x \right)
   \]
   Simplifying:
   \[
   f'(x) = 3e^x(1 + x)
   \]

4. **Find \(f'(1)\):**
   Substituting \(x = 1\) into the derivative:
   \[
   f'(1) = 3e^1(1 + 1) = 3e \cdot 2 = 6e
   \]

5. **Equation of the tangent line:**
   Now, we can write the equation of the tangent line using the point-slope form:
   \[
   y - 3e = 6e(x - 1)
   \]
   Expanding and simplifying:
   \[
   y = 6e(x - 1) + 3e = 6ex - 6e + 3e
   \]
   \[
   y = 6ex - 3e
   \]

### Final Answer:
The equation of the tangent line is:

\[
y = 6ex - 3e
\]

Would you like more details or further clarification?

---

Here are 5 follow-up questions to expand on this problem:
1. How does the product rule apply when differentiating functions like \(3xe^x\)?
2. What is the geometric interpretation of the tangent line to a curve at a point?
3. What happens to the slope of the tangent line as \(a\) changes for the function \(f(x) = 3xe^x\)?
4. How would you approach the problem if the function was more complex, like \(f(x) = x^2e^x\)?
5. How do the values of \(e\) and its derivatives influence the slope of the tangent line?

**Tip:** The product rule is a powerful tool when differentiating functions that involve a product of two functions, like \(x \cdot e^x\).

Find an equation for the line tangent to the graph of f(x) = 3xe^x at the point (a, f(a)) for a = 1.

Learn how to find the equation of the tangent line to the function f(x) = 3xe^x at the point where a = 1. This involves using the derivative via the product rule and applying the tangent line formula. Suitable for advanced high school or college calculus students.

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